Manoj and Shira were asked to find an explicit formula for the sequence $-9\,,-27\,,-81\,,-243,...$, where the first term should be $h(1)$. Manoj said the formula is $h(n)=-9\cdot3^{{n}}$, and Shira said the formula is $h(n)=-3\cdot3^{{n}}$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Manoj (Choice B) B Only Shira (Choice C) C Both Manoj and Shira (Choice D) D Neither Manoj nor Shira
Answer: In a geometric sequence, the ratio between successive terms is constant. This means that we can move from any term to the next one by multiplying by a constant value. Let's calculate this ratio over the first few terms: $\dfrac{-243}{-81}=\dfrac{-81}{-27}=\dfrac{-27}{-9}={3}$ We see that the constant ratio between successive terms is ${3}$. In other words, we can find any term by starting with the first term and multiplying by ${3}$ repeatedly until we get to the desired term. Let's look at the first few terms expressed as products: $n$ $1$ $2$ $3$ $4$ $h(n)$ ${-9}\cdot\!{3}^{0}$ ${-9}\cdot\!{3}^{1}$ ${-9}\cdot\!{3}^{2}$ ${-9}\cdot\!{3}^{3}$ We can see that every term is the product of the first term, ${-9}$, and a power of the constant ratio, ${3}$. Note that this power is always one less than the term number $n$. This is because the first term is the product of itself and plainly $1$, which is like taking the constant ratio to the zeroth power. Thus, we arrive at the following explicit formula (Note that ${-9}$ is the first term and ${3}$ is the constant ratio): $ h(n)={-9}\cdot{3}^{{\,n-1}}$ We can now expand this formula: $\begin{aligned} h(n)= &{-9}\cdot{3}^{{\,n-1}}\\\\ = & -9\cdot3^{{\,n}}\cdot 3^{-1}\\\\ = & -9\cdot \left(\dfrac{1}{3}\right)\cdot3^{{\,n}}\\\\ = &-3\cdot3^{{n}}\end{aligned}$ So Shira is definitely right. What about Manoj? We can see that $h(n)=-9\cdot3^{{\,n}}$ is not a correct formula, because the constant ratio is multiplied one extra time for each term. For instance, according to this formula, the value of the first term would be: $h(1)=-9\cdot3^{{\,1}} = -27$. However, according to our table of values, $h(1)=-9$. So Manoj is definitely wrong. Only Shira got a correct explicit formula.